Abstract

We introduce and study “isomonodromy” transformations of the matrix linear difference equation $Y(z+1)=A(z)Y(z)$ with polynomial $A(z)$. Our main result is construction of an isomonodromy action of $\mathbb{Z}^{m(n+1)-1}$ on the space of coefficients $A(z)$ (here $m$ is the size of matrices and $n$ is the degree of $A(z)$). The (birational) action of certain rank $n$ subgroups can be described by difference analogs of the classical Schlesinger equations, and we prove that for generic initial conditions these difference Schlesinger equations have a unique solution. We also show that both the classical Schlesinger equations and the Schlesinger transformations known in isomonodromy theory, can be obtained as limits of our action in two different limit regimes.

Similarly to the continuous case, for $m=n=2$ the difference Schlesinger equations and their $q$-analogs yield discrete Painlevé equations; examples include dPII, dPIV, dPV, and $q$-PVI.